Two counts, two growth stories
Double every edge of a cube. Do its surface area (the skin) and volume (the filling) double too?
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Step 1 of 4
- A small cube, edge e.
- Double every edge: a cube of edge 2e.
- Each face is four times bigger.
- The filling is eight times bigger.
The rule
No. The surface area grows 4 times and the volume 8 times.
A cube of edge 5 has surface area 150 and volume 125.
Double the edge to 10. Every face is 10 × 10 = 100. The surface area is 6 × 100 = 600. That is 150 × 4.
The volume: 10 × 10 × 10 = 1000. That is 125 × 8.
Let k be the number you multiply every edge by.
Multiply the edges by k. Surface area grows k² times. Volume grows k³ times.
Here e is the edge of the cube.
| Edge | Surface 6 × e × e | Volume e × e × e |
|---|---|---|
| 2 | 24 | 8 |
| 5 | 150 | 125 |
| 10 | 600 | 1000 |
| 15 | 1350 | 3375 |
Surface area is edge × edge for each face. Doubling the edge gives 2 × 2 = 4.
Volume is edge × edge × edge. Doubling the edge gives 2 × 2 × 2 = 8.